Apparent Impedance Analysis: A Small-Signal Method for Stability Analysis of Power Electronic-Based Systems

Abstract

Abstract-In this paper a new method for power system stability analysis is introduced. The method is based on injection of a small voltage or current in an arbitrary point of a power system. The apparent impedance is then defined as the ratio between the voltage and current at the injection point. It is shown that the apparent impedance can be used to estimate the eigenvalues of the system that are observable from the injection point. The eigenvalues are obtained by applying system identification techniques to the measured set of apparent impedances. The method is similar to the well established impedance-based stability analysis based on source and load impedance models. However, while the source/load impedance ratio is viewed as the minor-loop gain, the apparent impedance can be viewed as a closed-loop transfer function. It can also be expressed as the parallel connection of the source and load impedance. It is shown in the paper how the system eigenvalues can be extracted based on a set of apparent impedance values. The apparent impedance holds therefore complementary information compared with the existing impedance-based stability analysis. The method can also be used as a tool to validate analytically derived state-space models. In this paper the method is presented as a simulation tool, while further work will extend it to include experimental setups.Two case-studies are presented to illustrate the method. 1) A DC-case with a buck converter feeding a constant power load, and 2) a three-phase grid-connected VSC with current controller and PLL. The estimated (apparent) eigenvalues of the studied systems are equal to those obtained from the analytic state-space model.

Improving the Impedance-Based Stability Criterion by Using the Vector Fitting Method

“…Another problem of the IBSC that has been reported in [10]- [12] is that relative stability margins such as Phase Margin (PM) and Gain Margin (GM) are dependent on where the IBSC is considered. In [11], [12] it has been shown that the eigenvalues of the system are independent of the measurement point and they can be used as a relative stability measure.…”

“…Another problem of the IBSC that has been reported in [10]- [12] is that relative stability margins such as Phase Margin (PM) and Gain Margin (GM) are dependent on where the IBSC is considered. In [11], [12] it has been shown that the eigenvalues of the system are independent of the measurement point and they can be used as a relative stability measure. The eigenvalues can be obtained by an analytical evaluation of the whole system [13] or by approximating the driving point impedance, which is indeed the equivalent impedance of that node as a frequency response, into state equations using the Vector Fitting method [11], [12].…”

A dq-Domain Impedance Measurement Methodology for Three-Phase Converters in Distributed Energy Systems

“…The stability analysis of a power electronic system is usually carried out using small-signal models. The two types of small-signal methods are state-space-based and impedance-based analysis [13,14]. The advantage of state-space-based analysis is its ability to decompose the dynamics of the system into different oscillation modes, enabling access to the stability of each mode using eigenvalues.…”

“…The advantage of state-space-based analysis is its ability to decompose the dynamics of the system into different oscillation modes, enabling access to the stability of each mode using eigenvalues. However, for state-space models, complete knowledge about the system parameters is required [13]. Unfortunately, most of the electronic modules in modern DESs are from different vendors, and information about their design and parameters is often unavailable.…”

Bus Participation Factor Analysis for Harmonic Instability in Power Electronics Based Power Systems

“…Therefore, the formulation of the state matrices may become complicated in such large systems [21]- [23]. Apart from the state-space analysis, the impedancebased analysis approach is another powerful tool to predict the harmonic instability by calculating the ratio of the converter output impedance to the grid equivalent impedance at point of connection (PoC) of the converter [24]- [27]. The impedance-based approach has been applied in several applications, e.g., parallel LCL-filtered grid-connected converters [28], and voltagecontrolled and current-controlled inverters with LC-and LCL filters [11].…”